Montag begins to have a change of thinking when he starts to have conversations with Clarisse. She prompts him to question things and question himself. One quote, during the card game, which signifies a continuance of Montag's self examination (and a departure from his old way of thinking) is:
"I've tried to imagine," said Montag, "just how it would feel. I mean, to have firemen burn our houses and our books."
A key word in this quote is "imagine." Prior to Montag's awakening, he had gone through life passively, without questioning the status quo. The fact that he starts actively trying to imagine the effects of starting fires is significant. The fact that he voices it to Beatty and the other firemen is more significant because he's confronting the enforcers of the status quo.
Another quote that shows Montag has been examining his life and his society is when he asks if it is true that firemen used to put out fires. "Didn't firemen prevent fires rather than stoke them up and get them going?" At this point, Montag feels that Clarisse is speaking for him.
Both of these quotes reflect Montag's awakening. He never used to question anything about his life. Now, he's questioning his own happiness, the work of firemen, and his society in general.